It would probably be better to RP as a Sith who abandoned the Dark Side, rather than a Sith turned Jedi. The Jedi Order would be hard-pressed to accept such a member, due to how many times they get burned by Sith infiltrators. So, you could be a Light-side force user, maybe even fighting for the Republic, but you would have to do a LOT of impressive work to prove that you are no longer Sith and ready to join the Jedi Order.

(There will be many spoilers in this post for various entries in the Star Wars universe - comics, novels, video games, etc. For a much shorter, more direct answer to your questions, read the first and last paragraphs. Everything else is a history, as described in the second paragraph.)

A Pureblood Sith has never become a Jedi, because Pureblood Sith have never been raised in an environment where they would encounter any other Force philosophies than Dark Side and Sith perspectives. For most of the publishing history of Star Wars, Pureblood Sith have featured very rarely - they were shown during the Tales of the Jedi comic series, for example, but they had already been enslaved by former Jedi for a few hundred years by that point. They had never encountered the Jedi Order or Galactic Republic before those Dark Jedi conquered Korriban.

There are many examples of Jedi who have fallen to the Dark Side and ultimately returned to the Light Side, which in most cases is the closest you'll get to the question posed here. I'll start this with a history of the Jedi Order focused on conflicts between followers of the Light and Dark Sides of the Force, and will ultimately lead into examples of fallen Jedi who returned to the Order.

Around 24,500 BBY (Before the Battle of Yavin; Star Wars: Episove IV - A New Hope), the Jedi Order experienced its First Great Schism - war between adherents of the Light Side of the Force (Ashla), and those who chose to embrace the Dark Side (Bogan). This war resulted in the first Dark Jedi. Dark Jedi Xendor and his Legions of Lettow were allied with Arden Lyn and her Followers of Palawa; both were ultimately defeated by the Jedi Order.

Around 7000 BBY, the Jedi Order experiences its Second Great Schism, also known as the Hundred Year Darkness. After these new Dark Jedi were defeated, they retreated into the Unknown Regions of space - where they discovered a world known as Korriban, inhabited by a species called Sith. The exiled Dark Jedi conquered Korriban and enslaved the Sith, crowning themselves the Dark Lords of the Sith. The Dark Lords went on to conquer other planets, establishing the Sith Empire and the Sith Order.

Around 5000 BBY, two civilian explorers from Republic space, Jori and Gav Daragon, discovered Korriban while seeking out new hyperspace lanes for trade purposes. There, they were imprisoned by the Dark Lords who believed them to be spies for the Republic; while the Republic had all-but-forgotten the Sith, the Sith had never forgotten the Republic and the Jedi, even though they had lost the information necessary to return to that region of the galaxy. Naga Sadow used the Daragon's arrival as an opportunity to seize control of the Sith Council, and started the Great Hyperspace War.

Unknown to either at the time, Jori and Gav were both Force sensitive. Sadow took Daragon as his apprentice, though Daragon ultimately turned on his master. Sadow's fleet was forced into retreat after Daragon interrupted his concentration as he was using the Force to coordinate his forces (commonly called Battle Meditation). The Republic's forces gave chase and all but destroyed Sadow's fleet, and laid waste to Korriban itself. Sadow managed to retreat with a few ships to the jungle moon Yavin IV, where he would ultimately die. The Sith Empire and the Sith Order of this game are descendants of the original Empire and Order, specifically remnants that fled the Republic during the Great Hyperspace War.

Around 4400 BBY, a Jedi named Freedon Nadd decided to embrace the Dark Side in a pursuit of power, and slew his master. He made his way to the heart of the old Sith Empire, gathering knowledge of Sith and Dark Side ways, before ultimately discovering Naga Sadow's temples on Yavin IV. Sadow's spirit trained Nadd in the Sith ways; after Nadd felt he had learned all that he could from Sadow, he destroyed Sadow's spirit and conquered the world Onderon, declaring himself the Dark Lord of the Sith.

Around 4000 BBY, Jedi Master Arca Jeth, accompanied by his apprentices Ulic and Cay Qel-Droma, brought an end to Onderon's Beast Wars, which had technically raged since Nadd conquered the planet. They ended the influence of Nadd's lingering spirit and helped to restore peace to the planet.

Around 3997 BBY, a Sith-inspired cult known as the Krath developed in the Empress Teta system (the same system that was first invaded by Sadow's forces, and whose own military formed the vanguard of the Republic's response). Ulic Qel-Droma infiltrated the group, but ended up falling to the Dark Side and becoming the cult's leader. Around this same time, a Jedi Knight named Exar Kun discovered Freedon Nadd's tomb on Onderon's jungle moon Dxun; he followed in his master's footsteps by journeying to Yavin IV for more knowledge, then returned to Dxun to destroy Nadd's spirit. Kun initially went to the Krath cult to destroy it as well, seeing it and Qel-Droma as threats to his claim as Dark Lord of the Sith; the spirit of Marka Ragnos, a Dark Lord before Naga Sadow, was awoken, however, and named Kun the Dark Lord and Qel-Droma the apprentice. Kun and Qel-Droma's Krath cult then formed the Brotherhood of Darkness.

The Brotherhood of Darkness then launched the Great Sith War against the Jedi Order and the Republic. The war raged for four years, and ultimately ended with the capture of Qel-Droma and the retreat of Kun's forces to Yavin IV. Qel-Droma returned to the light, renouncing his Sith allegiances, and led the Republic's forces against Kun. Kun was killed on Yavin IV, though his spirit would live on in a temple that would later become home to Luke Skywalker's New Jedi Order.

Around 4250 BBY was the Second Great Schism, also known as the Vultar Cataclysm. This group of Dark Jedi ultimately destroyed themselves by attempting to use ancient alien technology against the Order and the Republic.

The Jedi Civil War took place between 3959-3956 BBY. This was the war chronicled in Knights of the Old Republic I, when Revan and Malak fell to the Dark Side, founded a new Sith Empire, and attempted to wrest control of the galaxy from the Republic. Revan, of course, returned to the Light Side after losing his memory, and went on to defeat Malak. Revan's "disappearance" into the Unknown Regions a couple years later is depicted in the novel The Old Republic: Revan by long-time BioWare writer Drew Karpyshyn. From KOTOR, you also know the stories of fallen Jedi such as Juhani and Yuthura Ban, both of whom returned to the Light due to Revan's influence. If you played KOTOR II: The Sith Lords, you also know the story of Atris, who fell to the Dark Side and chose exile for her crimes after being defeated by the Jedi Exile Meetra Surik.

Around 2000 BBY, the Fourth Great Schism in the Jedi Order occurred. Jedi Master Phanius fell to the Dark Side, declaring himself Darth Ruin of a new Sith Empire and launched the New Sith Wars. This war saw the ultimate demise of the Brotherhood of Darkness, with the final battle of the war occurring on the planet Ruusan. Darth Bane was the sole survivor, who decided to reform the Sith Order around a new doctrine - the Rule of Two.

Around 600 BBY, fallen Jedi Allya was banished to the planet Dathomir, where she founded a Dark Side cult known as the Witches of Dathomir (aka the Nightsisters).

Following the establishment of the Galactic Empire, Emperor Palpatine and Darth Vader conscripted many surviving members of the Jedi Order's AgriCorps (an organization to which many Padawans who could not pass the trials to become Knights were assigned) into becoming Dark Side Adepts. A Jedi Master Jerec survived the Jedi Purge by being out of touch with the Republic when Order 66 was enacted, and turned to the Dark Side upon his return to known space. In 5 ABY, he was killed by Jedi Knight Kyle Katarn. Katarn himself would later fall to the Dark Side, but returned to the Light and became a respected Jedi Master. One of Katarn's apprentices, Rosh Penin, would fall to the Dark Side but was redeemed by another of Katarn's apprentices, Jaden Korr. Korr himself would later come dangerously close to the Dark Side for his actions during the Second Galactic Civil War.

A human named Mara Jade was raised from an early age to act as the Emperor's Hand, a Dark Side assassin answerable only to Palpatine himself. Around 8-9 ABY (After Battle of Yavin), Mara Jade met Luke Skywalker, who was working to build his New Jedi Order. Mara Jade would ultimately train to become a Jedi Knight, would rise to the rank of Jedi Master, would marry Luke, and become mother to Ben Skywalker. She was later killed by a fallen Jedi.

Kam Solusar was apprenticed to his father Ranik Solusar during the Clone Wars, who was killed by Darth Vader during the Jedi Purge. Solusar spent years on the run but was ultimately captured; after extensive torturing, he fell to the Dark Side. Around 11 ABY, however, Luke Skywalker helped Solusar return to the Light. Solusar would become one of Luke's earliest students in the New Jedi Order, would marry fellow student Tionne, and would become a respected Jedi Master.

Around 11 BBY, Han Solo and Chewbacca met a young Force-sensitive man named Kyp Durron, who returned with them to Luke Skywalker's Jedi Praxeum on Yavin IV. Durron was seduced to the Dark Side and possessed by the spirit of Exar Kun, and used a superweapon known as the Sun Crusher to destroy the star of an Imperial system, killing millions of people. Kun's spirit was ultimately defeated and Durron returned to the Light. He would later become a Jedi Master.

A man named Flint, son of a Clone Wars-era Jedi named Flint Torul, served Darth Vader as a Shadow Hand during the Empire. He was also apprenticed to the Dark Lady of the Sith Lumiya, who was previously known as Shira Brie. Brie infiltrated the Rebel Alliance during the Galactic Civil War in an effort to assassinate Luke Skywalker; many years later, she would turn a different Jedi to the Dark Side during the Second Galactic Civil War. Flint, however, was killed by Lumiya in 14 ABY, when he turned back to the Light Side and refused to kill Skywalker.

Around 23 ABY, Jedi student Zekk fell to the Dark Side after being kidnapped by fallen Jedi Knight Brakiss. Brakiss had been a student of Luke Skywalker, but fell to the Dark Side and joined an organization called the Second Imperium; he founded the Shadow Academy to train a new generation of Dark Jedi. Zekk was later rescued and redeemed by his friends: Jaina and Jacen Solo (Han Solo and Leia Organa Solo's twin children) and Lowbacca (Chewbacca's nephew).

By the time of the Yuuzhan Vong War (25-29 ABY), there seemed to be very few Dark Jedi left in the galaxy - and there seemed to be no one claiming the Sith legacy (though Darth Krayt's One Sith was in hiding on Korriban, and would later wreak havoc on the galaxy around 140 ABY). Many Jedi would be tested greatly during that war, such as the aforementioned Jaden Korr. Anakin Solo, the third and youngest child of Han Solo and Leia Organa Solo, died during this war, as did the legendary Chewbacca.

Around 40 ABY, the Jedi Knight Jacen Solo came into contact with the Dark Lady of the Sith Lumiya. Solo became Lumiya's apprentice and took on the mantle of Darth Caedus. The Second Galactic Civil War was ongoing at this time between the Galactic Federation of Free Alliances and the Confederation, an alliance of planets led by Corellia that had seceded from the Alliance. As the commanding officer of the Galactic Alliance Guard - a special forces military unit created to deal with suspected secessionist and terrorist threats - Colonel Solo became highly influential in the GA government. With the assistance of GA Defense Force Supreme Commander Cha Niathal, he arranged for a bloodless coup in which Chief of State Cal Omas was removed from power, and the two military leaders were declared Co-Chiefs of State.

Over time, Jacen's parents, his sister Jaina, his uncle Luke and aunt Mara, and apprentice Ben Skywalker became concerned that Jacen was treading dangerously close to the Dark Side. Jacen killed Mara when she discovered that he had apprenticed himself to Lumiya, and before she was able to warn anyone. Ultimately, however, Jacen declared himself Darth Caedus to the entire Alliance and seized full power for himself. The Jedi Order had already withdrawn from the Alliance, establishing the Jedi Coalition with a secret group of military and political leaders as a government-in-exile. The Coalition would join with Admiral Niathal's official Alliance-in-Exile to defeat Caedus, who sought to reunite the galaxy under a vision similar to Palpatine's - except without the anti-alien bias. Jaina Solo killed her brother during the final battle of the war.

During the war, after Lumiya was killed by Luke Skywalker, Caedus used Jedi Knight Tahiri Veila's continued anguish over the death of Anakin Solo to turn her to the Dark Side as his apprentice. Tahiri ultimately returned to the Light Side, but not until after she had assassinated the Imperial Head of State on Caedus's orders. Her tale of redemption is still ongoing in the Fate of the Jedi novel series.

The FOTJ series - which still has one book left to release, in March 2012 - presents us with possibly the very first instance in which a Force user raised as Sith may renounce that allegiance and join the Jedi. Whether or not that character will is as yet unknown.

(I'm sure there are other examples that I've missed. One stands out to me right now from the list of fallen Jedi - Asajj Ventress, from the Clone Wars micro-series and CGI series, though she is not an example of someone who returned to the Light Side, so far as we currently know.)

This post will contain spoilers for a companion character in the Jedi Knight class story, in response to the post directly above:

I haven't gotten very far into the Jedi Knight class story yet, but if that's how it is presented when Lord Scourge joins the Knight, then the Knight is extremely naive. Scourge was a character in the Revan novel, and I'd be surprised to see him as potential candidate for redemption.

During that novel, he was absolutely devoted to the Sith Empire and the Sith Order - though he also became convinced that the Emperor presented an existential threat to the entire galaxy, and thus has to be killed.

Granted, that was 300 years ago compared to this game's timeline, but Karpyshyn depicted him as the type of zealot that I'd be very surprised to see change. Definitely interested to see where things go with him, though, when my Knight picks him up as a companion.

In the Jedi Knight class story Lord Praven (Sith Pureblood and one of Darth Angral's apprentices) can be redeemed and sent to the Jedi Council, where they accept him into the Jedi Order, and he's sent to help you later on Corellia.

Tyranny, like hell, is not easily conquered; yet we have this consolation with us, that the harder the conflict, the more glorious the triumph.
- T Paine

As an addendum to my previous, very long post, I just realized that I forgot possibly the most significant instance of a Jedi falling to the Dark Side only to be redeemed:

Luke Skywalker fell to the Dark Side during the Dark Empire comic series. In 10 ABY, armed with new superweapons, the Empire was striking back effectively at the New Republic. It was learned that the Empire's mysterious new leader was none other than Emperor Palpatine himself - who had been resurrected by using the Force to transfer his consciousness into a series of clone bodies. Skywalker believed that the only way to defeat Palpatine once and for all was to learn the ways of the Sith. However, like his father before him, Luke was seduced by the Dark Side and began fighting for the Empire. Ultimately, Luke was redeemed by Leia, and the two siblings fought together to kill Palpatine.

The Emperor returned again in Dark Empire II. Luke was continuing his quest to locate Force-sensitives who would join him in rebuilding the Jedi Order, and several of these individuals fought with Luke and the New Republic against the Empire during this series. Palpatine's ultimate and real defeat came in Empire's End, in which he attempted to transfer his consciousness to the body of Leia and Han's newborn son Anakin. Empatojayos Brand, a Jedi Knight and survivor of the Jedi Purge, instead "intercepted" the Emperor's consciousness and, as he died, took it with him to the Netherworld of the Force.

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The statements and opinions expressed on these websites are solely those of their respective authors and do not necessarily reflect the views, nor are they endorsed by Bioware, LucasArts, and its licensors do not guarantee the accuracy of, and are in no way responsible for any content on these websites.